(4b) An Approach to Model High Shear Wet Granulation by Using a Multi-Dimensional Population Balance Equation

High shear wet granulation is a key step in the manufacturing of tablets in the pharmaceutical industry. Granules possess major advantages compared to powders, which they are composed of, in terms of improved flow properties, reduced risk of segregation, increased homogeneity and lowered risk of dust explosion. From the pharmaceutical industry's point of view, there is a desire for predictive quantitative process models to be able to make in-silico process and scale-up simulations, which would shorten and reduce the cost of technology transfer from lab scale to industrial plant scale in drug development. Population balance models are shown to be adequate for describing nucleation and granule coalescence and breakage in the granulator and will ?one day be extremely valuable for scale-up in larger machines? (Faure et al., 2001).

In this study, a multi-dimensional population balance equation is applied to model the wet granulation process. The model is based on the concept described by Verkoeijen et al. (2002) but has been further developed. The concept is based on the fact that granule size, liquid saturation and porosity distributions can all be expressed as functions of three granule variables, i.e. the volume fractions of solid, liquid and air in a single granule.

This is not the most general statement of the multi-dimensional population balance equation, but it applies well to describe the wet granulation. In this study, the model is improved by a generalization that accounts for initial non-uniformly distributed liquid and air among the different particle size classes, but it is still assumed that granules with the same size have the same moisture content and porosity. For a single particle j, the particle properties such as the granule volume (v_j), porosity (ε_j), binder content (w_j) and pore saturation (S_j) can all be derived from the three volumes v_sj, v_lj and v_aj, where v_sj is the volume of solid material, v_lj is the volume of liquid and v_aj is the volume of air in the particle. The corresponding total volumes for all particles of class j are q_sj, q_ljand q_aj and the distributions of solid, liquid and air are written as q_s, q_l and q_a, respectively. The relation between v_ij and q_ij can be written as

                                                                                                                  (1)

where N_j is the number of particles of class j and i represents s, l or a. A detailed general derivation of the governing equations for the volume changing mechanisms, where no assumption about uniformly distributed liquid among the size classes has to be made, is made in this work. The resulting equations of change for q_sj, q_ljand q_aj due to coalescence can be written as

                                                                  (2)

                                                                   (3)

                                                                 (4)

where β_ij is the coalescence kernel. The kernel used is the kernel based on equipartition of fluctuating translational momentum between the colliding granules (similar to kinetic theory of granular flow) (Hounslow, 1998).

                                                                               (5)

In an earlier study performed by Jansson et al. (2005) this kernel has been found to be most appropriate for high shear granulation under present process conditions in a selection of different coalescence kernels.

In general, there is no restriction on the number of properties that can be defined, and therefore properties such as composition could also be included. As the nucleation mechanism in the beginning of the granulation process is complex and difficult to implement (Iveson et al., 2001), the focus of this study is solely on the wet massing period, where it is assumed that granule growth occurs due to coalescence only. Furthermore, an experimental exponential decay relation due to Iveson et al. (1996) is used for granule compaction:

                                                                                                                                    (6)

where k_c is the compaction rate constant and e_min is the minimal porosity that granules reach after a large number of collisions. These parameters are determined from experiments.

It is experimentally found that during the wet massing period the pores in the granules are fully saturated by liquid for the model system used, i.e. no air is present in the granules. A simplification of the developed model can therefore be used as no differential equation for the air present in the granules is needed. Particle volume distribution, liquid saturation, liquid to solid ratio and porosity of the granules can all be modelled, as these properties can all be expressed as combinations of three model parameters, i.e. the volume fraction of solid material, total liquid fraction and the liquid fraction inside the granules. The simulated results are compared to measurements from a series of five designed experiments where microcrystalline cellulose (MCC) is granulated and impeller speed and water content are varied. It is found that the evolution of the volume, liquid saturation and porosity distributions could all be explained by fitting the compaction rate (β₀) and coalescence rate (k_c). The figure below shows the simulated and measured distributions for one of the experiments. The figure contains the initial volume distribution, liquid saturation distribution and porosity distribution, and the corresponding measured and simulated distributions after 8 and 10 minutes of granulation. The strength of the model is that three distributions of independent variables are fitted using only two fitting parameters and still the general trends in the distributions are captured.

Simulation results for the experiment with 600 rpm impeller speed and 0.46 g added water/g dry material. β₀then becomes 19 s^-1 and k_c=0.0035 s^-1. Left column: Volume distribution, liquid saturation distribution and porosity distribution after 8 minutes of granulation. Right column: Volume distribution, liquid saturation distribution and porosity distribution after 10 minutes of granulation.

References

Faure, A., York, P., Rowe, R.C., (2001). Process control and scale-up of pharmaceutical wet granulation processes: a review. European Journal of Pharmaceutics and Biopharmaceutics, 52, 269-277.

Hounslow, M.J., (1998). The Population Balance as a Tool for Understanding Particle Rate Processes. Kona 16, 179-193.

Iveson, S., Litster, J., Ennis, B., (1996). Fundamental studies of granule consolidation. Part 1: Effects of binder content and binder viscosity. Powder Technology, 88, 15-20.

Jansson, A., Rasmuson, A., Niklasson-Björn, I., Folestad, S., (2004). High Shear Wet Granulation Modelling ? a mechanistic approach using population balances. Powder Technology. Accepted for publication.

Verkoeijen, D, Pouw, G.A., Meesters, G.M.H., Scarlett, B., (2002). Population balances for particulate processes ? a volume approach. Chemical Engineering Science, 57, 2287-2303.